The problem in the image reads:

Two bottles and their labels are mathematically similar.

• The smaller bottle contains 0.512 liters of water and has a label with an area of 96 cm².
• The larger bottle contains 1 liter of water.

Question:
Calculate the area of the larger label.

To solve this, we will use the fact that the bottles are similar in shape. For similar objects, the ratio of their surface areas is proportional to the square of the ratio of their volumes.

Step-by-step Solution:

1. Find the ratio of volumes:

   The volume of the smaller bottle = 0.512 liters.
   The volume of the larger bottle = 1 liter.

   The ratio of the volumes is: $\text{Volume Ratio} = \frac{1}{0.512} = 1.953125$

2. Area ratio for similar objects:

   Since the bottles are similar, the ratio of their areas will be the square root of the volume ratio: $\text{Area Ratio} = \sqrt{1.953125} \approx 1.3975$

3. Calculate the area of the larger label:

   The area of the smaller label is 96 cm². The area of the larger label is: $\text{Area of Larger Label} = 96 \times 1.3975 \approx 134.16 \, \text{cm}^2$

Thus, the area of the larger label is approximately 134.16 cm².

Would you like any further clarifications or details on any step?

Here are some related questions to deepen your understanding:

1. How would the solution change if the smaller bottle had a different volume?
2. What if the bottles had the same height but different diameters—would the same principles apply?
3. How can this concept be extended to calculate the surface areas of other similar 3D objects?
4. How is the formula for the area ratio of similar objects derived from the volume ratio?
5. What if the bottles were geometrically similar but not mathematically similar (e.g., one is wider)?

Tip: When dealing with similar figures, always remember that areas scale with the square of the linear dimensions, while volumes scale with the cube of the linear dimensions!